Global Dynamics and Stabilization of Zero-Mode Singularities in Multi-Scale Reaction-Diffusion Systems via Negative Coupling

This paper establishes a rigorous mathematical framework for the Multi-Scale Negative Coupled System (MNCS), a dynamical model describing hierarchical state spaces with directed, sign-structured interactions. We address the stabilization of reaction-diffusion systems on bounded domains $\Omega \subset \mathbb{R}^d$ ($d \le 3$) subject to homogeneous Neumann boundary conditions. A critical feature of this setting is the "zero-mode singularity," where the Laplacian operator possesses a trivial zero eigenvalue ($\lambda_0=0$), providing no linear dissipation for the spatial mean. We rigorously prove the global well-posedness of the system and the existence of a compact global attractor $\mathcal{A}$ in the phase space $\mathbb{H}=(L^2(\Omega))^N$. Utilizing the Moser-Alikakos iteration technique, we establish uniform $L^\infty(\Omega)$ bounds, overcoming the lack of Sobolev embedding from $H^1$ into $L^\infty$ in three dimensions. These bounds enable the derivation of explicit upper estimates for the fractal dimension of the attractor via the Kaplan-Yorke trace formula. We show that the dimension scales as $d_F(\mathcal{A}) \sim \max\{0, \mathcal{K}_{\mathcal{A}}-\gamma\}^{d/2}$, confirming that the negative coupling strength $\gamma$ acts as a global regularizer that compresses the phase space. The theoretical results are validated using a stiff-stable Second-Order Exponential Time Differencing (ETD2) scheme with Discrete Cosine Transform (DCT) to strictly enforce no-flux boundary conditions.